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Determine the characteristic polynomial of an algebraic 11 by 11 matrix?

Posted 2 years ago
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I wish to carry out a calculation which will produce a large volume of output. Specifically, to determine the characteristic polynomial of a general algebraic 11 by 11 symmetric matrix called 'm11' using Mathematica 10. I input the matrix then give the input CharacteristicPolynomial[m11, W] and for a while the banner at the top of the Mathematica says 'Running', but then it changes to just the file name without giving any output. Under 'Evaluation' the Abort Evaluation option now appears greyed-out, whereas it was in black to start. My computer runs Windows 10 and has 64 Gb or RAM and two 1 Tb hard discs. The smaller 10 by 10 matrix does calculate and gives an output, though it takes about 10 minutes. I appreciate that the 11 by 11 one will take longer. Mathematica does not give a clear indication that it has stopped calculating -- I cannot tell whether it is still working away in the background.
Therefore I cannot tell whether it will produce output in a few hours, or whether it has given up the job without telling me clearly. Can any of you experts please advise a novice on this, please?

4 Replies

You can use the Windows task manager, under processes to see if Mathematica's kernel is still working on the problem.

Just an observation: if a 9X9 matrix operation took 1 second and a 10X10 took 10 minutes, the 11X11 could take over 25 days!

Posted 2 years ago

Thank you for your helpful and prompt reply. Yes, Windows Task Manager is a good idea for monitoring the calculation. I have run it again and it started well, with the CPU usage increasing from 0% to about 10% over 1 minute while the Memory used grew steadily to about 18 GB (of 64 installed) over about 4 minutes. Over this time in Mathematica the Evaluation -> Abort Evaluation option was in black type. Then – quick as a flash – the CPU fell to 0 and the memory to only 132 MB. The Abort Evaluation option was then grey.

So it had died on me! I know 25 days is too long for any computer to countenance, but it should not give up after 4 minutes. The PC has been working OK otherwise.

For me there are now two questions : 1) why is my calculation suddenly ceasing, and without warning or explanation, and 2) would it be better if Mathematica itself provided some run-time information to the user on progress, rather than having to look into Windows Task Manager? Please advise.

You might try computing the LUDecomposition of m11-W*IdentityMatrix[11]. If that runs to completion, the determinant will be product of the main diagonal of the upper diagonal matrix. It may use a different method internally than CharacteristicPolynomial and, if so, would have a chance of running to completion.

Posted 1 year ago

Thanks again for suggestions. I have got round the problem by expanding the 11x11 matrix into its 11 10x10 cofactors and evaluating each of these individually, then combining their results. For my purposes I was eliminating selected terms on the way, so finished up with a maxtrix of manageable size. However, I could not do a 12x12 matrix by the same method.

The issue remains as to why Mathematica does not give the user some monitor of progress in large caculations. In my case it ran for a while, used a lot or memory, then suddently gave up and stopped with neither warning nor explanation. In contrast I have another maths algebra package which displays the CPU time used throughout any calculation.

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